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Mathematics > Combinatorics

arXiv:2202.06311 (math)
[Submitted on 13 Feb 2022]

Title:On maximum parallel classes in packings

Authors:Douglas R. Stinson, Ruizhong Wei
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Abstract:The integer $\beta (\rho, v, k)$ is defined to be the maximum number of blocks in any $(v, k)$-packing in which the maximum partial parallel class (or PPC) has size $\rho$. This problem was introduced and studied by Stinson for the case $k=3$. Here, we mainly consider the case $k = 4$ and we obtain some upper bounds and lower bounds on $\beta (\rho, v, 4)$. We also provide some explicit constructions of $(v,4)$-packings having a maximum PPC of a given size $\rho$. For small values of $\rho$, the number of blocks of the constructed packings are very close to the upper bounds on $\beta (\rho, v, 4)$. Some of our methods are extended to the cases $k > 4$.
Subjects: Combinatorics (math.CO)
MSC classes: 05B05
Cite as: arXiv:2202.06311 [math.CO]
  (or arXiv:2202.06311v1 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.2202.06311

Submission history

From: Douglas Stinson [view email]
[v1] Sun, 13 Feb 2022 13:41:11 UTC (14 KB)
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